The DNC challenge came after President Obama said earlier in the day that he will let Americans renew any health plans that do not meet his law’s coverage standards, attempting to quell the political firestorm — and consternation among many members of his own party — surrounding his broken promise that if people like their health insurance plan, they can keep it under the law.

“Has it been perfect? No,” Mr. Elleithee wrote. “President Obama and Democrats are the first to admit the website challenges have been unacceptable. Would more people be able to enroll if the website was working properly? Yes, of course. And they will.”

But, Mr. Elleithee continued, the prospect of the law actually working “terrifies” Republicans, and he wrote that their ultimate goal is to get rid of the law entirely.

Americans’ approval of the law has declined rapidly in recent weeks; 40 percent approve of it and 55 percent disapprove, according to new numbers from Gallup — the largest gap in a year.

Nevertheless, the DNC is trying to stay on the offensive in the fallout over troubles with the implementation of the law.

“I’m eager, anxious and proud to run on this in 2014,” Mr. Elleithee wrote. “Because every single Democratic member of Congress is committed to finding the solutions to make the Affordable Care Act work even better. The Republican Party? It shut down the federal government, delivering a $24 billion hit to our economy.”

“I’ll take that fight any day of the week,” he continued. “Republicans tried to ‘tattoo’ Obamacare to our foreheads in 2012. We won. They tried it in Virginia in 2013. We won. Now they want to try it again in 2014? OK. I’ll buy the ink.”

Republicans didn’t appear to be frightened. The National Republican Senatorial Committee (NRSC) re-tweeted a tweet from a DNC spokeswoman highlighting Mr. Elleithee’s post.

The National Republican Congressional Committee, which oversees the House Republican campaign for 2014 also sent out a tweet changing a DNC slogan from “Change that matters” to “Change that health plan.”